While many literature have already studied the impact of global matrix conditioning [1]}, [2]}, [3]}, our work is the first to study how local matrix distribution impact the convergence speed. In order to focus on the impact of local data structure, we first introduce the following assumption on the global data matrix \(\) . We assume the global covariance matrix \(^T\)  has fixed conditioning number. Specifically, we assume that the global covariance matrix has fixed largest and smallest eigenvalues. We further assume that the global covariance matrix is normalized by its Frobenius norm, hence the largest eigenvalue of the global covariance matrix is smaller than 1. Since the global matrix conditioning would influence the convergence speed, fixing the global matrix conditioning allows us to purely focus on the effect on how data share influences convergence. Any data share across local centers would not influence the conditioning of the global covariance matrix, however, it still significantly influence the convergence speed.
